Two numbers are in the ratio 3 : 4 . If their LCM is 240 , the smaller of the two number is

Options

A. 100
B. 80
C. 60
D. 50

Answer

To find the smaller of the two numbers given their ratio and LCM, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Ratio**:
The two numbers are in the ratio 3:4. We can represent these numbers as:
- First number = 3x
- Second number = 4x
Here, 'x' is a common factor.

2. **Identify the LCM**:
We are given that the LCM of these two numbers is 240.

3. **Use the Relationship Between LCM and HCF**:
The relationship between the two numbers, their LCM, and their HCF (Highest Common Factor) is given by the formula:
\[ \text{LCM} \times \text{HCF} = \text{Product of the two numbers} \]
Therefore, we can write:
\[ LCM = 240 \]
\[ \text{Product of the two numbers} = (3x) \times (4x) = 12x^2 \]

4. **Set Up the Equation**:
From the relationship, we have:
\[ 240 \times \text{HCF} = 12x^2 \]
Since the HCF of the two numbers is 'x', we can substitute:
\[ 240 \times x = 12x^2 \]

5. **Simplify the Equation**:
Divide both sides by 'x' (assuming x ≠ 0):
\[ 240 = 12x \]

6. **Solve for x**:
Now, divide both sides by 12:
\[ x = \frac{240}{12} = 20 \]

7. **Find the Smaller Number**:
Now that we have the value of 'x', we can find the smaller number:
\[ \text{Smaller number} = 3x = 3 \times 20 = 60 \]

8. **Conclusion**:
The smaller of the two numbers is **60**.